What does it mean to "have a multiplicative inverse of modulo 10!"?

Question

Here's the question:

What's the smallest integer > 1 that has a multiplicative inverse modulo 10! (that is, 10 factorial)?

What does that mean?

I understand that:

We say that x is the multiplicative inverse of a modulo N whenever ax = 1(mod N)

But... we only know N in this case right? I'm very confused

Answer 1

You have the right definition. And it is a theorem that $a$ has a multiplicative inverse modulo $m$ if and only if $a$ and $m$ are relatively prime. So what is the smallest integer $a\gt 1$ such that $a$ and $10!$ are relatively prime?

Answer 2

Hint $\ $ By Bezout, $\rm\ a\ $ is invertible mod $\rm\:n!\:$ iff $\rm\ a\ $ is coprime to $\rm\:n!.\:$ Thus if $\rm\ a>1\:$ then every prime divisor of $\rm\ a\ $ is $\rm\, > n,\:$ e.g. the next prime $\rm\,> n.\:$